Modelling of Thin Elastic Plates with Small Piezoelectric Inclusions and Distributed Electronic Circuits. Models for Inclusions that Are Small with Respect to the Thickness of the

Journal of Elasticity55: 111–141, 1999.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

111

Modelling of Thin Elastic Plates with SmallPiezoelectric Inclusions and Distributed ElectronicCircuits. Models for Inclusions that Are Small withRespect to the Thickness of the Plate

ÉRIC CANON and MICHEL LENCZNERLaboratoire de Mathématiques de Besançon UMR 6623, Équipe de Calcul Scientifique, Universitéde Franche-Comté, 16 Route de Gray, F-25030 Besançon Cédex, FranceE-mail: [email protected]

Received 11 March 1999; in revised form 11 October 1999

Abstract. This paper is devoted to the modelling of thin elastic plates with small, periodicallydistributed, piezoelectric inclusions, in view of active controlled structure design. The initial equa-tions are those of linear elasticity coupled with the electrostatic equation. Different kinds of boundaryconditions on the upper faces of inclusions are considered, corresponding to different ways of control:Dirichlet, Neumann, local or nonlocal mixed conditions. We compute effective models when thethicknessa of the plate, the characteristic dimensionε of the inclusions, andε/a tend together tozero. Other situations will be considered in two forthcoming papers.

Mathematics Subject Classifications (1991): 35B27 homogenization of partial differential equa-tions in media with periodic structures, 73B27 nonhomogeneous materials and homogenization.

Key words: linear elasticity, piezoelectricity, homogenization, plate theory, composite materials,prescribed electric potential, local electric circuits, nonlocal electrical circuits, transfinite networks,smart materials.

1. Introduction

1.1. GENERAL

This paper is part of systematic work devoted to the derivation of effective modelsfor piezoelectric/elastic composite plates including elementary electronic circuits.In [4],we considered three dimensional elastic plates with a small number of piezo-electric inclusions, and we derived effective models when the thickness of the platetends to zero. The models are static (and linear) but may be extended to dynamicvia the Laplace transform.

In the present paper, we consider plates with a great number of piezoelectrictransducers, periodically distributed in an elastic matrix, requiring homogenization.

112 E. CANON AND M. LENCZNER

Two small parameters are thus involved in our analysis: the thicknessa of the plateand the characteristic dimensionε of inclusions.Effective modelsmean that wecompute the limit models whena andε simultaneously tend to zero. The fact thata andε tend together to zero ensures that all the possible limit models are obtained.Three different situations actually occur according to whethera/ε→ 0, ε/a→ 0or ε = a. The aim of the present paper is to obtain models in the case where theinclusions are small with respect to the thickness of the plate, that isε/a→ 0. Thetwo other situations will be treated in two forthcoming papers. We remark that(simplified) models fora/ε→ 0 were presented in [5].

The goal we have in mind is to control structures by electrical regulation appliedto the upper and lower faces of piezoelectric transducers. More precisely, we try toconceive distributed electronic circuits which act on structures for the purpose ofcontrol. As in [4], we consider different possibilities for the boundary conditions onthe upper faces of inclusions, corresponding to different kinds of control: Dirichletconditions, if the tension is controlled, Neumann conditions, if the current is con-trolled, and mixed conditions, if inclusions are connected to R-L-C circuits. In thislast class, we consider the case where the upper and lower faces of each inclusionare connected, and the case where, in addition, each inclusion is connected to itsdirect neighbours. From a mathematical point of view, this corresponds to localmixed conditions and to nonlocal mixed conditions, respectively.

For the model associated with nonlocal boundary conditions, a Laplace operatorin the in-plane direction of the plate arises, acting on the transverse componentL0

3of the electric field. This is a model of transfinite network type, as described, forexample, in Zemanian [12, 13] with a different approach. Let us mention thatone may choose a priori the form of the operator onL0

3 (and the correspondingboundary conditions) by appropriately connecting the inclusions to each other (andto the outside of the domain). In fact, we get here a complete family of transfinitenetworks. This seems particularly interesting in the perspective of building relevantcontrollers.

To derive the effective models, we use a mixing of two-scale convergence [1, 10]and of classical arguments of plate theory [6, 7, 10]. Note, however, that, as in [4],the derivation is made in the space of the gradients of solutions. This seems tobe unusual, but allows a more synthetic and readable presentation of the modelsthemselves as well as of their derivation. We think that this formalism, in itself, isan interesting contribution of this work.

The obtained models do have rather a simple structure. The effective modelfor Dirichlet conditions has the same form as the purely elastic plate model; theinfluence of piezoelectrics only appears in the definition of the effective coefficientsand as a source term on the right hand side. This is not the case for nonlocal mixedconditions: because of the differential operator induced by the R-L-C circuits, acoupling arises between mechanical effects and a transverse component of theelectric field. For local mixed conditions, the situation is intermediate: see thecomments at the end of Section 5.

COMPOSITE PIEZOELECTRIC PLATES WITH ELECTRONIC 113

For problems which include homogenization and plate theory, one must men-tion the work of Caillerie [3]. Caillerie treated the case of thin static elastic plateswith rapidly oscillating coefficients, using the energy method of Tartar [2, 11]. Itshould be noted that the parametersa andε tend (except fora = ε) successivelyand independently to zero in [3].

A more extensive bibliographical review on piezoelectric plate models is givenin [4].

1.2. DETAILED CONTENTS

Section 2 is devoted to the presentation of the initial 3-dimensional equations:elasticity and piezoelectricity equations in their linear and static versions. Thepiezoelectric inclusions are assumed to be strictly included in the elastic matrix,which is considered to be electrically insulated. For simplicity, as it is usuallythe case in applications, the stiffness, piezoelectricity and permitivity tensors areassumed to be constant in the direction of thickness of the plate. In the same spirit,the upper and lower faces of the piezoelectrics are assumed to be metallized, that is,covered with a thin film of conductive metal. Concerning the equation of elasticity,standard boundary conditions are considered: Neumann conditions on the upperand lower faces of the plate, Neumann and Dirichlet conditions on the lateralboundary. For the equation of piezoelectricity, we consider Neumann conditionson the lateral boundary, Dirichlet condition on the lower faces. As mentionedin Section 1.1, various boundary conditions are considered on the upper faces,namely: Dirichlet, Neumann, local and nonlocal mixed conditions. These kinds ofconditions are, to our knowledge (except Neumann conditions), unusual in platetheory. They thus constitute an interesting point of this paper.

The corresponding weak formulations are presented in Section 3. In the sequel,because of the relative formal complexity of the models, and because we want totreat the various boundary conditions together, as much as possible, we adopt syn-thetic tensorial notation rather than fully expanded formulae. We strongly believethat this allows a better description of our computations as well as of our limitmodels.

The precise assumptions on the data are presented in Section 4.1. We give, inparticular, the correct scalings. From a practical point of view, this indicates howelectric circuits must be chosen to obtain a significant influence on the effectivebehaviour of the material. Resulting a priori estimates and first convergence resultsare given in Sections 4.2 and 4.3.

Section 5 is devoted to the statement of the main result of the paper, Theo-rem 5.1: an effective 2-dimensional plate model for each type of electrical bound-ary condition, when the inclusions are much smaller than the thickness of theplate.

Theorem 5.1 is proved in Section 6 by lettinga, ε, ε/a tend simultaneously to 0in the weak formulations of Section 3. The proof is in three steps.


The first one, which is mathematically most difficult, consists in characterizingtwo-scale limits of the strains and of the electric field. These results are new, even inthe case of pure elasticity. Caillerie considered weak limits only; the intermediatetwo-scale limits were not described in [3]. These results are of general interest andmay apply to various situations which concern homogenization and plate theory.

The second step consists in eliminating the local variabley by computing themicroscopic fields (depending ony) with respect to the macroscopic fields (de-pending only on the macroscopic variablex). Here, we use the classical argumentsof linear homogenization.

The third step consists in eliminating (part of) the transverse components of thestrains and of the electric field, which may be computed with respect to the othercomponents of these fields. This elimination slightly departs from the classicalplate theory, because of the nonstandard boundary conditions on the faces of theinclusions.

We use the same formalism as in [4], based on tensorial notation and products,and on simple algebraic operations such as projections. It allows us to deal rela-tively easily with complex computations. Completely explicit formulae would belengthy and limit the readability. In our approach, steps 2 and 3 are almost formalcomputation and may be easily adapted to variants of our models. In this way, onecould also easily extend the results to multilayered plate models, as in [4].

To conclude the paper, in Section 7, we propose, an illustration of our mod-elling. We consider a transversally isotropic material (a PZT ceramic, for instance),with Dirichlet conditions. This is the simplest possible example, because the effectof piezoelectricity does not occur in the isotropic material. We use the program-ming package Mathematica to compute, from the general formulation of Theo-rem 5.1, quite explicit formulae for the effective coefficients. By comparing thecompact formulae of Section 5 with the expanded formulae of Section 7, one mayappreciate the formalism used in [4] and in the present paper. Moreover, the use ofMathematica shows that this formalism is not only elegant; it is also practical.

2. Equations of 3-dimensional Piezoelectricity

This section is devoted to the presentation of the initial 3D equations. The cor-responding weak formulations are given in Section 3, the corresponding effectivemodels are calculated in Sections 4–6.

2.1. GEOMETRY

Let a be the positive parameter measuring the thickness of the plate. The 3-dimen-sional plate is initially represented by�a = ω×]−a, a[, ω being a bounded do-main ofR2. (Using the classical change of scales and variables introduced in [6],we shall in fact work on the fixed domain� = ω×]−1,1[.)


Figure 1. Composite plate with piezoelectric inclusions.

Figure 2. Elementary cell with piezoelectric inclusion and metallization.

Let ε > 0 denote the characteristic dimension of inclusions. The domainω isdivided into two subdomainsωε1 andωε2 that are constructed as follows. LetYbe a rectangle subdomain ofR2 such that, without loss of generality,|Y | = 1.Let Y1 ⊂⊂ Y with |Y1| > 0, andY2 = Y \ Y1. The setωε1 is a union of all theεY -periodic translations ofεY1 that are strictly contained inω, whileωε2 = ω \ωε1.Let b = (a, ε). The elastic matrix is represented by�b2 = ωε2×]−a, a[, the set ofall piezoelectric inclusions by�b1 = ωε1×]−a, a[.

The inclusions are numbered by a multi-indexi = (i1, i2) ∈ Iε. Then, 〈·〉idenotes the mean value on the upper face of the inclusion indexed byi. For everyfunctionψ on�a, ψi is the restriction ofψ to the inclusioni.

The boundary ofω is assumed to be smooth and divided into two regular partsγD andγN , with |γD| > 0. The boundary of� is divided into:0aD = γD×]−a, a[,0aN = (γN×]−a, a[) ∪ (ω × {−a, a}). The boundary of�b1 is divided into0b+1 =ωε1× {a}, 0b−1 = ωε1× {−a} and0b1 = ∂ωε1×]−a, a[.

The current point in�a is xa = (x1, x2, xa3), wherexa3 ∈]−a, a[ and x =

(x1, x2) ∈ ω. The current point inY is y = (y1, y2). The derivatives with respectto xα, xa3 andyα are denoted by∂α, ∂3 and∂yα , respectively. The outer unit normalto the boundaries of�a andY is denoted byn andnY , respectively.


Specifically for the scaled domain�, a constant use is made of

M(f ) = 1

2

∫ 1

−1f (x3)dx3 and N (f ) = f −M(f ) for f ∈ L1(−1,1). (1)

Finally, let us mention that when referring to the fixed domain�, the geometricnotation is the same, the subscripta being removed, if necessary.

2.2. OTHER NOTATIONS

Bold characters are used for vector and matrix valued functions and, possibly,for the corresponding functional spaces. We constantly use Einstein’s conventionof summation on repeated indices, with summation from one to three for Latinindices, from one to two for Greek indices.

2.3. EQUATIONS OF3-DIMENSIONAL PIEZOELECTRICITY

The mechanical displacementsub = (ubi )i=1,2,3 and the electrical potentialϕb aregoverned by the linear equations of piezoelectricity in their static version. In thissection, we recall these equations that underlie our models. The boundary condi-tions for the upper and lower faces of inclusions that characterize different modelsare specified in 2.4.

The plate is submitted to the volume mechanical forcesfb = (f bi )i=1,2,3 in �a,and to the surface mechanical forcesgb = (gbi )i=1,2,3 on0aN .

For anyv ∈ H1(�a), let us denote the strains by

sij (v) = 1

2(∂ivj + ∂jvi) ∀i, j ∈ {1,2,3}, ∀v ∈ H1

(�a). (2)

The stressesσ b = (σ bij )i,j=1,2,3 and the electrical displacementsDb= (Dbi )i=1,2,3

are then given by{σ bij = Rεijklskl

(ub)+ dεkij ∂kϕb in �,

Dbk = −dεkij sij

(ub)+ cεki∂iϕb in �b1.

(3)

The mechanical equilibrium equations and the mechanical boundary conditionsare:

−∂jσ bij = f bi in �, σ bij nj = gbi on0aN for i = 1,2,3,

(4)ub = 0 on0aD.

The electrostatic equation and the electrical boundary conditions on the lateralfaces of the inclusions are:

−∂iDbi = 0 in�b1, Db · n = 0 on0b1. (5)


In (3), Rε = (Rεijkl)i,j,k,l=1,2,3, dε = (dεkij )i,j,k=1,2,3 andcε = (cεij )i,j=1,2,3 denote thetensors of stiffness, piezoelectricity, and permitivity. They satisfy

Rεijkl = Rεklij = Rεjikl, cεij = cεji , dεijk = dεikj ∀i, j, k, l ∈ {1,2,3}. (6)

We assume that the piezoelectric inclusions are electrically insulated from theelastic matrix. The electrical influence of�b2 on�b1 is, therefore, neglected in ouranalysis. Though, it is convenient to define the tensorsRε, dε, cε on the wholedomain�a. We let, therefore,

cεij = 0, dεijk = 0 in�b2 ∀i, j, k ∈ {1,2,3}. (7)

For the electrostatic equation, we go now into detail about the boundary conditionson the upper and lower faces of inclusions.

2.4. BOUNDARY CONDITIONS ON THE UPPER AND LOWER FACES OF THE

PIEZOELECTRIC INCLUSIONS

For the sake of conciseness, we only consider situations where all the faces aremetallized, as it is usually the case in applications. From a mathematical point ofview, this means that the electric field is constant on each face of each inclusion.Considering nonmetallized faces would lead to unnecessary technical complica-tions. However, let us note that nonmetallized faces were considered by the authorsin [4] for models with few inclusions.

Three kinds of conditions are considered:

2.4.1. Dirichlet Conditions

ϕb ={ϕbm + aϕbc on0b+1 ,ϕbm − aϕbc on0b−1 ,

(8)

whereϕbm andϕbc are constant on each inclusion.This condition may result from the connection of each piezoelectric to the out-

put of a tension source providing tensionϕbc , or to the input of a current amplifier(hereϕbc = 0). In both cases, one of the faces is connected to the ground equalto ϕbm.

2.4.2. Neumann and Local Mixed Conditions

ϕb = ϕbm on0b−1 , 〈Db · n〉i = −Gaϕb + hb on0b+1 ∀i ∈ Iε, (9)

whereϕb = ϕ|0b+1 − ϕbm. The functionsϕbm andhb are constant on each inclusion,G is a fixed nonnegative constant.

Equation (9) covers two sorts of boundary conditions. IfG = 0, (9) is a Neu-mann condition

〈Db · n〉i = hb on0b+1 ,


which arises when the inclusions are connected to the output of a current sourcehb,or to the input of a tension amplifier (hb = 0). WhenG > 0, (9) is the true mixedcondition. It occurs when the upper and lower faces of each inclusion are connectedby an R-L-C circuit of impedancea/G, hb being an additional source of current.

REMARK 2.1. The above explanations are slightly inaccurate. In fact, the currentwhich flows out of an inclusion is the time derivative of〈Db · n〉i. One may thinkof (9) as the Laplace transform of the Kirchoff law.

2.4.3. Nonlocal Mixed Conditions

They occur when dielectric inclusions are also connected together by R-L-C cir-cuits. We consider here the case where the upper face of each inclusion is connectedwith each of its direct neighbours, but not to the outside of the plate. This isdescribed as follows.

Let us introduce the shift operators

T 1+1:

Iε → N2,

i 7→ (i1+ 1, i2),T 2+1:

Iε → N2,

i 7→ (i1, i2+ 1),

T 1−1:

Iε → N2,

i 7→ (i1− 1, i2),T 2−1:

Iε → N2,

i 7→ (i1, i2− 1).

With the convention

ϕbT α−1(i)− ϕbi = 0 if T α−1(i) /∈ Iε,

(10)ϕbT α+1(i)

− ϕbi = 0 if T α+1(i) /∈ Iε, for α = 1,2,

the Kirchhoff law leads here to〈Db · n〉i =

2∑α=1

G1

aε2

(ϕbT α−1(i)

− 2ϕbi + ϕbT α+1(i)

)− Gaϕbi + hb, ∀i ∈ Iε, on0b+1 ,

ϕb = ϕbm on0b−1 .

(11)

Figure 3. Cell with nonlocal electric circuit.


Hereaε2/G1 (G1 > 0) designates the common impedance of the circuits linkingtwo adjacent inclusions.

REMARK 2.2. Condition (11) clearly corresponds to a discrete Laplace operatorin the two directions of the plate here with discrete homogeneous Neumann condi-tions (10). Due to the above particular scaling on the impedance, in the asymptoticprocess(a, ε)→ (0,0), this generates a Laplace operator on the transverse compo-nent of the electric field. It is worth emphasizing that one can choose in advance theoperator (and the corresponding boundary conditions) on the transverse componentof the electric field by appropriately choosing the way to connect the upper facesto each other.

2.4.4. General Comments

In the sequel, we often use common formulations for the above three boundaryconditions. To do so, we need to definehb, ϕbc , G, andG1 for all the models withthe conventions

hb = 0 for Dirichlet conditions, ϕbc = 0 for mixed conditions, (12)G andG1 are two given nonnegative constants,

G = G1 = 0 for Dirichlet conditions,G1 = 0 for local mixed conditions,GG1 > 0 for nonlocal mixed conditions.

(13)

Unlike [4], we do not treat separately the case of Neumann conditions. From amathematical point of view, it does not differ from the case of local mixed con-ditions. One simply has to setG = 0 in the local mixed condition to obtain thecorresponding model.

Since all the faces are metallized,ϕb, ϕbm, andϕbc are constant on each face ofeach inclusion. Also, because the current is provided by a single wire, the sameproperty holds forhb.

The relevance of the scaling(G/a)−1 and(G1/(aε2))−1 on the capacities will

become apparent in the next sections. It will clearly indicate, according to inclu-sions and thickness, the type of circuit that must be chosen to obtain a significanteffect on the global behaviour of the plate.

3. Weak Formulations

Notation, equations, and boundary conditions were stated in the previous sections.The aim of the present section is the formulation, on the fixed domain�, of thecorresponding weak formulations. The effective models are deduced from theseweak formulations (18) in the next two sections.


3.1. SCALING OF THE EQUATIONS

Using the standard change of variablesxa → x = t (x1, x2, x3) = t (x1, x2, xa3/a),

equations of Section 2 are reformulated on� = ω×]−1,1[. As already men-tioned, the geometrical notation for the domain� is the same as for�a, the indexa being removed when necessary. The corresponding scaling for volume forces,surface forces, and displacements is classical [6]:

ub(x) = (ub1(xa), ub2(xa), aub3(xa)) in �,

fb(x) = (f b1 (xa), f b2 (xa), a−1f b3(xa))

in �,gb(x) = (gb1(xa), gb2(xa), a−1gb3

(xa))

onγN×]−1,1[,gb(x) = a−1

(gb1(xa), gb2

(xa), a−1gb3

(xa))

onω× {−1,1}.The current sourceshb, the electric potentialsϕb, ϕbm, andϕbc are unchanged. As

in the sequel we only work on the reference domain�, we use again, for simplicity,the notationub, fb, gb, hb, ϕb, ϕbm, andϕc, without hats.

For V = (v,ψ) ∈ H1(�)× H 1(�ε1), we define the scaled strain tensor and thescaled electric fieldK a(v) = (Ka

ij (v))i,j=1,2,3 andL a(ϕ) = (Lai (ϕ))i=1,2,3 byKaαβ(v) = sαβ(v) for α, β = 1,2,

Ka3α(v) = Ka

α3(v) = a−1sα3(v) for α = 1,2,Ka

33(v) = a−2s33(v),Laα(ϕ) = ∂αϕ for α = 1,2 andLa3(ϕ) = a−1∂3ϕ,

(14)

where∂3 represents now∂/∂x3. We also use the global notation

M a(V) = t((Kaαβ(v)

)α,β=1,2,

(Kaα3(v)

)α=1,2,K

a33(v),

(Laα(ψ)

)α=1,2, L

a3(ψ)

). (15)

3.2. WEAK FORMULATION

We put together the tensorsRε, dε, andcε in a global stiffness-piezoelectricity-permitivity tensorRε, which is the 10× 10 symmetric matrix written in a formatcompatible with (15):(Rεαβγ δ)α,β,γ,δ=1,2 (2Rε

αβγ3)α,β,γ=1,2 (Rεαβ33)α,β=1,2 (dεγ αβ)α,β,γ=1,2 (dε3αβ)αβ=1,2

(2Rεα3γ δ)α,γ,δ=1,2 (4Rε

α3γ3)α,γ=1,2 (2Rεα333)α=1,2 (2dε

γα3)α,γ=1,2 (2dε3α3)α=1,2

(Rε33γ δ)γ,δ=1,2 (2Rε33γ3)γ=1,2 Rε3333 (dεγ33)γ=1,2 dε333

(−dεαγ δ)α,γ,δ=1,2 (−2dεαγ3)α,γ=1,2 (−dε

α33)α=1,2 (cεαγ )α,γ=1,2 (cεα3)α=1,2

(−dε3γ δ)γ,δ=1,2 (−2dε3γ3)3,γ=1,2 −dε333 (cε3γ )γ=1,2 cε33

.

(16)

The linear forms associated with the mechanical and electric loads are

lbu(v) =∫�

f bi vi dx +∫0N

gbi vi ds, and lbϕ(L3) = ∫

�ε1

hbL3 dx.


Given the assumption on metallization, the set of admissible electric potentialsis chosen by

H 1c

(�ε1) = {ψ ∈ H 1(�ε1);ψ is constant in each connected part of0ε+1 ∪ 0ε−1

}.

(17)

The Hilbert spacesWε andWbD are defined by

(i) Dirichlet conditions:Wb

D = Wε(ϕbm, ϕ

bc

):= {

(v, ϕ) ∈ H1(�)×H 1c

(�ε1); v = 0 on0D,

ϕ = ϕbm ± aϕbc on0ε±1},

Wε = Wε(0,0).

(ii) Mixed conditions:

WbD =Wε

(ϕbm) := {(v, ϕ) ∈ H1(�)×H 1

c

(�ε1); v = 0 on0D,

ϕ = ϕbm on0ε−1},

Wε =Wε(0).

The backward difference operator∇εx

is defined inclusion by inclusion by(∇εxψ)i = ε−1 t(ψi − ψT 1−1(i)

, ψi − ψT 2−1(i)

) ∀i ∈ Iε ∀ψ ∈ H 1c

(�ε1).

The weak formulations on the scaled domain� for the coupled prob-lems (3)–(5) and (8)–(11), with the conventions (7), (10), (12), (13), are thensummarized by:

∫�tM a(V)RεM a

(Ub)dx + 2

∫�ε1GM

(La3(ϕb))

M(La3(ψ)

)dx

+ 2∫�ε1G1∇εxM

(La3(ϕb)) · ∇ε

xM(La3(ψ)

)dx = lbu(v)+ lbϕ

(La3(ψ)

)∀V = (v, ψ) ∈Wε, with Ub = (ub, ϕb) ∈Wb

D.

(18)

The mean operatorM is defined in (1). We used (6) to reorganize the first term andthe relations

ϕb|0ε+1 = ϕb

|0ε+1 − ϕb

|0ε−1 = 2M(∂3ϕ

b)

and

ψ|0ε+1 = 2M(∂3ψ) for the other terms.

REMARK 3.1. It is worth pointing out that, for mixed conditions, the connectionof the two faces of each inclusion introduces the mean value of the transversecomponent of the electric field in the equations.


In (18) and throughout the paper, the products between matrices have to beunderstood as bloc matrices products, where in each bloc Einstein’s convention ofsummation is used. For example,RεM a(Ub) is equal to

(Rεαβγ δK

aγ δ(u

b)+ 2Rεαβγ3K

aγ3(u

b)+ Rεαβ33K

a33(u

b)+ dεγαβLaγ (ϕb)+ dε3αβLa3(ϕb))α,β=1,2(

2Rεα3γ δK

aγ δ(u

b)+ 4Rεα3γ3K

aγ3(u

b)+ 2Rεα333K

a33(u

b)+ 2dεγα3L

aγ (ϕ

b)+ 2dε3α3La3(ϕ

b))α,β=1,2

Rε33γ δKaγ δ(u

b)+ 2Rε33γ3Kaγ3(u

b)+ Rε3333Ka33(u

b)+ dεγ33L

aγ (ϕ

b)+ dε333La3(ϕ

b)(−dεαγ δKaγ δ(ub)− 2dε

αγ3Kaγ3(u

b)− dεα33K

a33(u

b)+ cεαγ Laγ (ϕb)+ cεα3La3(ϕ

b))α=1,2

−dε3γ δKaγ δ(ub)− 2dε3γ3Kaγ3(u

b)− dε333Ka33(u

b)+ cε3γ Laγ (ϕb)+ cε33La3(ϕ

b)

.

4. Assumptions on the Data. A Priori Estimates. Convergences

The aim of this section is twofold. The detailed assumptions on the data are statedin 4.1. The resulting a priori estimates and first convergence results are given in 4.2.

4.1. ASSUMPTIONS ON THE DATA

We use in this paper the notion of two-scale convergence of Allaire [1] and Nguent-seng [9]. Since we also need two-scale convergence for functions defined on�ε1,we use the following practical definition.

DEFINITION 4.1. A sequence(ψb) of L2(�ε1) is said to two-scale converge to alimit ψ in L2(� × Y1) if ψ ∈ L2(� × Y1) and if (P εψb) two-scale converges toPψ in L2(� × Y ), whereP ε andP denote the extension by 0 from�ε1 to� andfrom�× Y1 to�× Y , respectively.

In addition to the standard symmetry assumptions (6), the tensorsRε, dε, andcε

constituting the stiffness-piezoelectricity-permitivity tensorRε are assumed to sat-isfy

(Rε)

two-scale converges inL2(�× Y ) to some limitR ∈ L∞(�× Y ),‖Rε‖L∞(�) 6 C, Rε does not depend onx3,

limε→0

∥∥Rε∥∥

L2(�)= ‖R‖L2(�×Y),

tKR εK > c‖K‖2 ∀K ∈ R9 with Kij = Kji, a.e. inω,tLcεL > c‖L‖2 ∀L ∈ R3, a.e. inωε1.

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Here and throughout the paper,c andC designate generic positive constants, notdepending ona andε.

REMARK 4.1. In view of the symmetry relationsdεijk = dεikj , coercivity for cε

andRε implies coercivity forRε. Conversely, two-scale convergence forRε im-plies two-scale convergence forRε, cε, anddε. The corresponding limits are natu-rally denoted byR, c, andd.


The mechanical forces are assumed to satisfy fb ∈ L2(�),gb ∈ H1/2(0N),

(fb) converges weakly inL2(�) to some limitf,(gb) converges weakly inL2(0N) to some limitg.

(20)

The assumptions relative to the electrical boundary conditions are:hb, ϕbm, andϕbc are constant on each inclusion,(hb) two-scale converges inL2(ω× Y1) to some limith ∈ L2(ω),

(ϕbm) two-scale converges inL2(ω× Y1) to some limitϕm ∈ H 1(ω),

(ϕbc ) two-scale converges inL2(ω× Y1) to some limitϕc ∈ L2(ω).

(21)

REMARK 4.2. Sinceϕbc , ϕbm, andhb are constant on each inclusion, their two-

scale limits do not depend ony in Y1.Let us recall that the convention (12)–(13) have been chosen to definehb, ϕbc ,

G, andG1 for all the models.

4.2. A PRIORI ESTIMATES. CONVERGENCES

Let us introduce the space of Kirchhoff–Love’s displacement fields

VKL ={v ∈ H1(�); v = 0 on0D, (si3(v))i=1,2,3 = 0

},

or equivalently,

VKL ={t (v1− x3∂1v3, v2 − x3∂2v3, v3); v1, v2 ∈ H 1(ω), v3 ∈ H 2(ω),

v1 = v2 = v3 = 0 on0D}.

In the sequel, forv ∈ VKL, we frequently use the practical notationv = t (v1, v2).A priori estimates and the resulting convergence results for the sequence(ub, ϕb)

are summarized in the following lemma. The convergence statements hold a priorifor a subsequence. However, since we see with hindsight (from uniqueness of thesolution to the limit problem) that the complete sequences converge, we omit tomention the extractions of subsequences.

LEMMA 4.1. If assumptions(6), (19)–(21)and conventions(12), (13)hold, thenfor sufficiently smallb:

(i) for each fixedb there is a unique solution to problem(18);(ii) ‖K a(ub)‖L2(�) + ‖L a(ϕb)‖L2(�ε1)

+G1‖∇εxM(La3(ϕb))‖L2(�ε1)

6 C;(iii) there existsM = (K ,L ) ∈ (L2(�×Y ))7×(L2(�×Y1))

3 such that(M a(ub))two-scale converges toM in L2(�× Y )× L2(�× Y1);

(iv) there existsu ∈ VKL andu1 = t (u11, u

12,0) with u1

1, u12 ∈ L2(�;H 1

] (Y )/R),such that(ub) converges weakly tou in H1(�), (∇xub) and(∂3ub) two-scaleconverge to∇xu+∇yu1 and∂3u, respectively, inL2(�× Y );


(v) (ϕb) two-scale converges toϕm in L2(�× Y1);(vi) there existsϕ1 ∈ L2(�;H 1(Y1)) such thatt (L1, L2) = ∇yϕ1;(vii) M(L3) is independent ofy, and for Dirichlet conditionsM(L3) = ϕc;

(viii) In the case of nonlocal mixed conditions,M(L3) ∈ H 1(ω) and(∇ε

xM(La3(ϕ

b))) two-scale converge to∇xM(L3) in L2(�× Y1).

Proof. Point (i) is a direct application of Lax–Milgram’s lemma. Point (ii) isobtained with standard arguments by choosing(v, ψ) = (ub, ϕb − (ϕbm + ax3ϕ

bc ))

in the case of Dirichlet conditions,(v, ψ) = (ub, ϕb−ϕbm), otherwise. Point (iii) isa direct consequence of (ii).

Let us prove (iv). First, as (K a(ub)) is bounded inL2(�), Korn’s inequality im-plies that(ub) is bounded inH1(�). Then, from [1, Proposition 1.14], there existsu ∈ H1(�) andu1 = t (u1

1, u12, u

13) ∈ L2(�;H1

](Y )/R), such that (after extractionof a subsequence, if necessary)(ub) converges weakly tou in H1(�), (∇xub) two-scale converges to∇xu + ∇yu1 in L2(� × Y ), (∂3ub) two-scale converges to∂3uin L2(� × Y ). Now, the sequences(Ka

i3(ub)) being bounded, in view of (14), the

sequences(si3(ub)) strongly tend to 0. As they also converge to(si3(u)), this provesu ∈ VKL.

To complete the proof of (iv), it remains for us to show thatu13 may be taken

as equal to zero. We re-use the fact that the sequences(sα3(ub)) strongly convergeto 0. Strong convergence implies two-scale convergence, and thus,sα3(u)+∂yαu1

3 =∂yαu

13 = 0. Then, becauseu1

3 is defined up to the function ofx, we are free to chooseu1

3 = 0.To prove (v) we note that(ϕb) is bounded inL2(�ε1) (see (ii)). Thus,(ϕb)

two-scale converges to some limitϕ in L2(� × Y1). As (La3(ϕb)) = (a−1∂3ϕ

b) isbounded,ϕ does not depend onx3. Similarly, as the quantities(Laα(ϕ

b)) = (∂αϕb)are bounded,ϕ does not depend ony. Hence,ϕ ∈ L2(ω). Now, to computeϕ, weonly need to pass to the limit in∫�ε1

∂3ϕbψ(x3 − 1)dx = −

∫�ε1

ϕbψ dx + 2∫ωε1

(ϕbm − aϕbc

)ψ dx ∀ψ ∈ H 1(ω).

The left-hand side tends to 0 because(La3(ϕb)) is bounded and thereforeϕ = ϕm.

To prove (vi), we use the identity∫�ε1

∇xϕb · ψεdx = −∫�ε1

ϕb(divxψ)εdx −

∫�ε1

ϕb

ε(divyψ)

εdx

+∫0ε1

ϕbψε · n dσ ε ∀ψ ∈ D(�× Y ),

whereψε denotes the functionx 7→ ψ(x, x/ε). We chooseψ such that divyψ = 0in �× Y1 andψ · nY = 0 in�× ∂Y1, and pass to the limit asb→ 0. With (v), weget ∫

�×Y1

(t (L1, L2)−∇xϕm

) · ψ dxdy = 0.


This proves thatt (L1, L2)− ∇xϕm is a gradient with respect toy. Remarking that∇xϕm = ∇y(y · ∇xϕm), this proves thatt (L1, L2) is a gradient with respect toy.

The first part of (vii) is obtained by remarking thatM(La3(ϕb)) = a−1M(∂3ϕ

b)

= a−1(ϕb|0ε+ −ϕb|0ε−) is constant on each inclusion (becauseϕb ∈ H 1c (�

ε1)). Hence,

its two-scale limit does not depend ony.For Dirichlet conditions, (8) also implies thatM(La3(ϕ

b)) = ϕbc . Hence, passingto the limit:M(L3) = ϕc.

Let us prove (viii). Letξ designate the two-scale limit of(∇εxM(La3(ϕ

b))).Let ψ ∈ D(� × Y1). For ε small enough,ψε(x) = ψ(x, x/ε) vanishes in allnoninternal inclusions. Then, the following integration by parts formula holds:

ε−1∫�ε1

∑i∈Iε

M(La3(ϕbi − ϕbT α−1(i)

))ψε

i dx

= ε−1∫�ε1

∑i∈Iε

M(La3(ϕbi))(ψε

i − ψεT α+1(i)

)dx

for α = 1,2. Passing to the limit, asψ is regular, this yields∫�×Y1

ξαψ dxdy = −∫�×Y1

M(L3)∂αψ dx, α = 1,2.

This proves thatM(L3) ∈ H1(ω) andξα = ∂αM(L3). 2

5. Main Result: Limit Models

This section is devoted to the presentation of the effective 2-dimensional platemodels. They are obtained by lettinga, ε, ε/a tend simultaneously to 0 in (18).As a unique asymptotic situation arises here, we subsequently know that the samemodels would be obtained by deriving the first 3-dimensional homogenized equa-tions by lettingε tend to 0 (a fixed), and then applying the asymptotic method inthe plate theory asa→ 0.

The derivation of limit models is made up of three steps. The first one, themost difficult mathematically, consists in characterizing the limits(K ,L ) definedin Lemma 4.1. The second step consists, as usual, in linear homogenization, ineliminating the local variabley. This is realized by computing the microscopicfields (depending ony) with respect to the macroscopic fields (depending onlyonx). The third step consists in eliminating (part of) the transverse components ofthe fields (homogenized strains and the electric field) that we compute with respectto the other components. This elimination differs from the classical plate theorybecause of the nonstandard boundary conditions on the faces of inclusions.

The notation related to step 2 is presented in Section 5.1. The notation relatedto step 3 is presented in Section 5.2. The effective models are then summarized byTheorem 5.1, in Section 5.3. The proof of Theorem 5.1 is postponed to Section 6.


Let us stress again that our approach allows a synthetic and readable presen-tation of our results. Compare, for example, the notation of Section 5.2 below,with the expanded expression presented in Section 7 for a transversally isotropicmaterial. Our notation is also practical: the calculations of Section 7 (based on thegeneral notation of Section 5) have been worked out using Mathematica.

5.1. NOTATION RELATED TO HOMOGENIZATION

Similarly to (2), let

Sαβ(v) = 1

2(∂yαvβ + ∂yβvα), Sα3(v) = 1

2∂yαv3, α, β = 1,2. (22)

Let us define

Z = {(1,1), (1,2), (2,1), (2,2), (1,3), (2,3), (3,3), 3}.The local variables(ui , ϕi) ∈ (H 1

] (Y ))3×H 1(Y1), needed to compute the homoge-

nized elasticity-piezoelectricity-permitivity tensorRH , are defined, for eachi ∈ Z,as the solutions of the local problems:∫

Y

(Sαβ(v), Sα3(v3), ∂yαψ)

Rαβγ δ 2Rαβγ3 dγαβ2Rα3γ δ 4Rα3γ3 2dγα3

−dαγ δ −2dαγ3 cαγ

Sγ δ(ui)Sγ3(u

i3)

∂yγ ϕi

dy

=∫Y

(Sαβ(v), Sα3(v), ∂yαψ)

Rαβγ δ 2Rαβγ3 Rαβ33 dγαβ d3αβ

2Rα3γ δ 4Rα3γ3 2Rα333 2dγα3 2d3α3

−dαγ δ −2dαγ3 −dα33 cαγ cα3

(23)

×

δi,γ δ0γδi,33

δi,γδi,3

dy ∀(v, ψ) ∈ (H 1] (Y )

)3×H 1(Y1),

whereδi,j is the Kronecker symbol fori, j ∈ Z.The tensorL, stored in a format compatible withR, is defined as

L =

(Sαβ(uµρ))α,β,µ,ρ=1,2(Sα3(uµρ))α,µ,ρ=1,2

02×2(∂yα ϕ

µρ)α,µ,ρ=1,202×2

(Sαβ(uµ3))α,β,µ=1,2(Sα3(uµ3))α,µ=1,2

02(∂yα ϕ

µ3)α,µ=1,202

(Sαβ(u33))α,β=1,2(Sα3(u33))α=1,2

0(∂yα ϕ

33)α=1,20

02×2×202×202

02×202

(Sαβ(u3))α,β=1,2(Sα3(u3))α=1,2

0(∂yα ϕ

3)α=1,20

.

The homogenized stiffness-piezoelectricity-permitivity coefficients are then givenby

RH =∫Y

(Id + tL

)R(Id +L)dy. (24)


5.2. NOTATION RELATED TO PLATE THEORY

The notation is:

5 and51 are the projections from(L2(�))10 onto its subspaces of the formt(04, (Ki3)i=1,2,3 and02, L3

), t (09, L3), respectively, 52 = 5−51,

TN = −(5RH5

)−15RH,

TM = −(52R

H52)−152R

H for Dirichlet and nonlocal conditions,TM = −(5RH5+ 2G51)

−15RH for local mixed conditions,RN =

(Id + tTN

)RH(Id + TN ),

RM =(Id + tTM

)(RH + 2G51

)(Id + TM),

RMixM = |Y1|

(tTM −

(Id + tTM

)(RH + 2|Y1|G51

)× (5RH5+ 2|Y1|G51

)−1).

(25)

The notation in (25) is not completely correct. The inverted matrices are not in factinvertible as applications from(L2(�))10 to (L2(�))10, but on the relevant sub-spaces. For example, the inversion of5RH5 is meant for the restricted application5(L2(�))10 7→ 5(L2(�))10. In practice,(5RH5)−1 is obtained by deleting thezero lines and columns of5RH5, inverting the resulting matrix and incorporatingthe results in the right place in a 10×10 matrix of format (16). A detailed exampleis given in Section 7.

REMARK 5.1. MatricesRM, RN , andRMixM have the same format (16) asR.

The corresponding submatrices are naturally denoted by

RM,dM, cM,RN ,dN , cN ,RMixM ,dMix

M , andcMixM .

Because of the projections, these matrices are sparse matrices. Only the coeffi-cientsRMαβγ δ,RN αβγ δ, dM3αβ , dM3αβ , cM33, anddMix

M3αβ are needed in the followingmodels.

5.3. MODELS

Let

l(v) =

∫�fivi dx +

∫0Ngivi ds − 2

∫ω

sαβ(v)dM3αβϕc dx

for Dirichlet conditions,∫�fivi dx +

∫0Ngivi ds + 2

∫ω

sαβ(v)dMixM3αβhdx

for local mixed conditions,∫�fivi dx +

∫0Ngivi ds

for nonlocal mixed conditions.


THEOREM 5.1. Assume that the hypothesis of Lemma4.1holds. Assume thata,ε, andε/a tend to zero. Then:

(i) in the case of Dirichlet or local mixed electrical boundary conditions, thesequence(ub) converges tou = t (u1− x3∂1u3, u2− x3∂2u3, u3) ∈ VKL whichis the unique solution of:∫

ω

(2sαβ(v)RMαβγ δsγ δ(u)+ 2

3∂2αβv3RN αβγ δ∂

2γ δu3

)dx = l(v) ∀v ∈ VKL;

(ii) in the case of nonlocal mixed electrical boundary conditions, the sequence(ub,M(La3(ϕ

b))) converges to(u, L03) ∈ VKL × H 1(ω) which is the unique

solution to:∫ω

(2(sαβ(v), L3

) ( RMαβγ δ dM3αβ

eM3γ δ cM33+ 2|Y1|G)(

sγ δ(u)L0

3

)+ 2

3∂2αβv3RN αβγ δ∂

2γ δu3

)dx + 4|Y1|

∫ω

G1∂αL3∂αL03 dx

= l(v)+ 2∫ω

L3hdx ∀(v, L3) ∈ VKL ×H 1(ω).

5.3.1. Comments

• All the models are independent ofϕm. Only the difference of potential betweenthe upper and lower faces does influence the effective behaviour of the plate.• In both cases, equations foru3 and u are uncoupled. This would, however, no

longer be the case for multilayered plates. See [4].• For Dirichlet and local mixed conditions, the limit model has the standard form

of a two-dimensional elastic plate. The influence of inclusions only appears inthe definition of the effective coefficients, and as a source term on the right-handside.• For nonlocal conditions, the situation is more interesting. The coupling arises

between the mechanical effects and the transverse electric field induced by theinclusions. The form of the differential operator (here, a Laplace operator) actingonL0

3 depends only of the choice of connections between inclusions. However,given that in (ii) the equations foru3 on the one hand, for(u1, u2, L

30) on the

other hand, are uncoupled, the transverse displacement control would requirethe consideration of multilayered plates, as in [4].• Formulation (ii) is more general than formulation (i). First, for time dependent

problems, even for local mixed conditions, this formulation could be appliedbecauseG could be a combination of time derivatives which cannot be simplyinverted. Second, also for local conditions, when thinking in terms of control,one may prefer to consider model (ii) (withG1 = 0) rather than model (i). Therole ofG is more apparent in (ii): the local mixed conditions actually correspondto the operator onL0

3 without derivatives.


6. Proof of Theorem 5.1

6.1. STEP1: CHARACTERIZATION OF THE LIMIT M

6.1.1. Some Notation

We give here some notation that allows a more elegant presentation of the results.Let C∞] (Y ) denote the subspace ofY -periodic functions ofC∞(R2). For any

functionv ∈ D(�,C∞] (Y )), we systematically denote byvε ∈ D(�) the functionx 7→ v(x, x/ε). A similar convention is used for functions ofD(�,C∞] (Y1)).

In what follows, we consider in (18) two-scale admissible test functions in

W1ad =

{V1 = (v1, ψ1) ∈ (D(

�,C∞] (Y ))3×D

(�,C∞] (Y1)

));v1 = 0 on0D × Y

},

and test functions in

Wad ={(v, ψ) ∈ H1(�)×9ad(�); v = 0 on0D

},

where

9ad(�) = D(]−1,1] × ω) for mixed conditions,9ad(�) = D(]−1,1[×ω) for Dirichlet conditions.

ForV ∈W1ad, we introduce the key decomposition ofM a(Vε):

M a(Vε) = (

M00(V))ε + 1

ε

(M10(V)

)ε + 1

a

(M01(V)

)ε+ 1

aε

(M11(V)

)ε + 1

a2

(M02(V)

)ε, (26)

and we recall that forV ∈Wad :

M a(V) = M00(V)+ 1

aM01(V)+ 1

a2M02(V), (27)

where, with definition (2) forSαi, asV = (v, ψ):

M00(V) = t((sαβ(v)

)α,β=1,2,03, (∂αψ)α=1,2,0

),

M10(V) = t((Sαβ(v)

)α,β=1,2,03, (∂yαψ)α=1,2,0

),

M01(V) = t(02×2,

(sα3(v)

)α=1,2,04, ∂3ψ

),

M11(V) = t(02×2,

(Sα3(v)

)α=1,2,04

),

M02(V) = t(02×2,02, s33(v),03

).

(28)

Associated subspacesM,M−2,M−1 andM0 of (L2(�×Y ))7× (L2(�×Y1))3 are

defined by


M−2 = {t (02×2,02,K33,03);K33 ∈ L2(�)},

M−1 = {t(02×2, (kα3/2)α=1,2,03, L3)+M11

(V2); kα3, L3 ∈ L2(�),

v23 ∈ L2

(�;H 1

] (Y )), with M(L3) = 0 for Dirichlet conditions

},

M0 = {M00(v,0)+M10(V1);

v ∈ VKL,V1 ∈ L2(�;H1

](Y ))2× {0}× ∈ L2

(�;H 1(Y1)

)},

M =M−2⊕M−1⊕M0.

REMARK 6.1. EachM ∈M is associated with(v, v1, ψ) ∈ VKL×L2(�;H1

](Y ))

× L2(�;H 1(Y1)), wherev1 = t (v11, v

12, v

23).

6.1.2. Three Preliminary Lemmas

The first two lemmas are density results that allow us to pass from admissible testfunctions to test functions inM,M−2,M−1, andM0. Lemma 6.1 deals with mixedconditions. For Dirichlet conditions, each function of9ad(�) is trivially identifiedto a function ofH 1

c (�ε1). This is no longer the case for mixed conditions (see

definition (17) ofH 1c (�

ε1)). It is the aim of Lemma 6.1 to overcome this difficulty.

LEMMA 6.1. For mixed conditions, for eachψ ∈ 9ad(�), there exists a sequence(ψε) with ψε

|�ε1 ∈ H 1c (�

ε1) such that(∂3ψ

ε) strongly converges to∂3ψ in L2(�).Proof. Let ωεi denote the mean section of the inclusion numberi. To obtain

Lemma 6.1, we simply need to chooseψε defined by

ψε = 1

|ωεi |∫ωεi

ψ(x)dx in ωεi ∀i ∈ Iε. 2LEMMA 6.2.

(i) The set{M02(V);V ∈Wad} is dense inM−2,(ii) The set{M01(V)+M11(V1);V ∈Wad,V1 ∈W1

ad, v3 = 0} is dense inM−1,(iii) The set{M00(V)+M10(V1);V ∈ VKL × {0},V1 ∈W1

ad} is dense inM0.

Proof. Point (i) follows, for instance, from the density of{∂3v3; v3 ∈ D(ω×]−1,1])} in L2(�). Point (ii) is similar. For Dirichlet conditions, we remark thatthe density of{∂3ψ;ψ ∈ D(ω×] − 1,1[)} is in {L3 ∈ L2(�);M(L3) = 0} only.Point (iii) is straightforward. 2LEMMA 6.3. Let (uε) be a bounded sequence inH 1(�). Let u ∈ H 1(�) andu1 ∈ L2(�;H 1

] (Y )) be functions such that(uε) weakly converges tou in H 1(�),(∇uε) two-scale converges to∇u+∇yu1 in L2(�× Y ). Then

limε→0

∫�

uε

ε∂yαv

ε dx =∫�×Y

u1∂yαv dxdy ∀v ∈ D(�;C∞] (Y )

).


Proof.We simply need to pass to the limit in∫�

∂αuεvε dx = −

∫�

uε(∂αv)ε dx −

∫�

uε

ε

(∂yαv

)εdx.

The integration by parts of the first term on the right-hand side then yields theresult. 2

6.1.3. Characterization ofM

Defineφc by φc = t (09, ϕc) for Dirichlet conditions,φc = 010 for mixed condi-tions.

LEMMA 6.4. Assume that assumptions of Lemma4.1hold. Assume thata, ε, andε/a tend to zero. Then(M a(Ub)) two-scale converges to

M = t

((sαβ(u)+ Sαβ

(u1))

α,β=1,2,1

2

(kα3+ ∂yαu2

3

)α=1,2,K33,

(∂yαϕ

1)α=1,2, L3

)∈ φc +M

which is the unique solution of∫�×Y

tMRM dxdy + 2G∫�×Y1

M(L3)M(L3)dxdy

+ 2G1

∫�×Y1

∂αM(L3)∂αM(L3)dxdy = lu(v)+ lϕ

(L3) ∀M ∈M, (29)

where

lu(v) =∫�

fivi dx +∫0Ngivi dx,

(30)lϕ(L3) =

∫�×Y1

hL3 dxdy = |Y1|∫�

hL3 dx.

Proof. The proof is in two steps. We first establish thatM satisfies the weakformulation (29). We then show thatM ∈ φc +M. Uniqueness of the solution of(29) is a simple consequence of Lax–Milgram’s lemma.

In the case of Dirichlet conditions, we chooseV ∈Wad as a test function in theweak formulation (18). We then multiply bya2, a, and 1 successively and pass tothe limit in each case. With definitions (26), (27), and (28) we, thus, get

∫�×Y

tM02(V)RM dxdy = 0 ∀V ∈Wad,∫�×Y

tM01(V)RM dxdy + 2∫�×Y1

GM(L3)M(∂3ψ)dxdy

+2∫�×Y1

G1∂αM(L3)∂αM(ψ)dxdy = lϕ(∂3ψ)

∀V ∈Wad with M02(V) = 0,∫�×Y

tM00(V)RM dxdy = lu(v) ∀V ∈Wad

with M02(V) = M01(V) = 0.

(31)


For mixed conditions, (31) also holds, but we need to start in (18) withψε as inLemma 6.1 instead ofψ .

Choose nowV1ε: x 7→ V1(x, x/ε), whereV1 ∈W1ad, as a test function in (18).

With definition (26), multiplication byaε andε yields∫�×Y

tM11(V1)RM dxdy = 0 ∀V1 ∈W1

ad,∫�×Y

tM10(V1)RM dxdy = 0 ∀V1 ∈W1

ad,

with M11(V) = M02(V) = 0.

(32)

Now, using Lemma 6.2, point (i), the first equation in (31) is equivalent to the weakformulation (29) withM−2 instead ofM. Also, using Lemma 6.2 (ii), the secondequation in (31) (withv3 = 0) and the first equation in (32) are equivalent to (29)withM−1 instead ofM. Last, using Lemma 6.2 (iii), the third equation in (31) (withv3 ∈ VKL, that ensuresM02(V) = M01(V) = 0) and the second equation in (32)(with v1

3 = 0) are equivalent to (29) withM0 instead ofM. As (29) holds for anyM in M−2,M−1, andM0, it holds inM = M−2⊕M−1⊕M0. This ends the proofof the first part of Lemma 6.4.

Now we prove thatM ∈ φc +M, or in other words, thatM has the form asannounced in the lemma.

The formKαβ = sαβ(u) + Sαβ(u1) for α, β = 1,2 is a direct consequence ofLemma 4.1, point (iv) and of definition (14):Ka

αβ(vb) = sαβ(vb). The formLα =

∂αϕ1 is proved in Lemma 4.1, point (vi).ConcerningK33, we simply need to show that it is independent ofy. To do this,

we pass to the limit in the identity

1

a2

∫�

∂3ub3

(ε(∂β∂yαv)

ε + (∂2yαyβ

v)ε)dx

= 2ε

a

∫�

Kaβ3

(uε)(∂yα∂3v)

εdx + ε2

a2

∫�

1

εubβ∂yα∂

233v

εdx

which holds forα, β ∈ {1,2} andv ∈ D(�;C∞] (Y )) (recall that this implies that∂yαv is Y-periodic). The first term on the right-hand side tends to zero because(Ka

β3(uε)) is bounded. Using Lemma 6.3, the second term on the right-hand side

also tends to zero. Hence,∫�×Y

K33∂2yβyα

v dxdy = 0 ∀v ∈ D(�;C∞] (Y )

).

This proves thatK33 does not depend ony.For (K13,K23), first note, using a few integrations by parts, that∫�

(∂αu

bβ − ∂βubα

)∂3v

εdx

= 2a∫�

Kaβ3

(ub)(∂αv

ε + 1

ε∂yαv

ε

)dx − 2a

∫�

Kaα3

(ub)(∂βv

ε + 1

ε∂yβv

ε

)dx

∀v ∈ D(�;C∞] (Y )

). (33)


By multiplying (33) byε/a and passing to the limit, one gets∫�×Y

(Kβ3∂yαv −Kα3∂yβv)dxdy = 0 ∀v ∈ D(�;C∞] (Y )

).

Hence,t (K13,K23) is curl-free with respect toy. Thus, there exist(k13, k23) ∈L2(�) andu1

3 ∈ L2(�;H 1] (Y )) such that 2Kα3 = kα3 + ∂yαu1

3 (see [8, Section3], if necessary for this well-known orthogonality result in the context of periodicfunctions).

To complete the proof, it remains for us to examineL3. Passing to the limit in

ε

∫�ε1

La3(ϕb)((∂αψ)

ε + 1

ε(∂yαψ)

ε

)dx = ε

a

∫�ε1

Laα(ϕb)∂3ψ

εdx, α = 1,2,

as the quantitiesLai (ϕb) are bounded, one gets∫

�ε1

L3∂yαψ dxdy = 0, α = 1,2.

This proves thatL3 does not depend ony and thus completes the proof of Lemma 6.1if the mixed conditions case. To conclude for Dirichlet conditions, it suffices toremember thatM(L3) = ϕc (see Lemma 4.1 (vii)). Hence,M(L3− ϕc) = 0. 2

6.2. STEP2: HOMOGENIZATION

The weak formulation (29) being established, the next step consists in eliminatingthe local variabley. This requires the auxiliary functions(uγ δ,uγ3,u33,u3) definedin (23). We use the decomposition

M = M x +M11(U1)+M10(U1)in (29), where

M x = t((sαβ(u)

)α,β=1,2, (kα3/2)α=1,2,K33,02, L3

),

U1 = t(u1

1, u12, u

23, ϕ

1).

LEMMA 6.5. LetM be the solution of(29). Then,

M11(U1)+M10(U1) = LM x,

and(u, (kα3)α=1,2,K33, L3) ∈ VKL × (L2(�))4 is the unique solution of∫�

(tMRHM x + 2|Y1|GL3M(L3)+ 2|Y1|G1∂αM(L3)∂αL3

)dx

= lu(v)+ lϕ(L3) ∀v ∈ VKL, ∀((kα3)α=1,2, K33, L3

) ∈ L2(�). (34)


The linear formslu andlϕ are defined in (30).Proof. In (29), we choose test functionsM ∈M of the formM = M11(V1) +

M10(V1), where

V1 = t(v1

1, v12, v

23, ψ

1) ∈ L2

(�;H1

](Y ))× L2

(�;H 1(Y1)

).

We obtain∫Y

t(M11(V1)+M10(V1))R(M11(U1)+M10(U1))dy= −

∫Y

t(M11

(V1)+M10

(V1))

R dyM x

∀V1 = t(v1

1, v12, v

23, ψ

1) ∈ H1

](Y )×H 1(Y1), almost everywhere in�.

Hence,U1 is the unique solution (up to a function ofx) to the above variationalproblem. But asM x is independent ofy, one may chooseU1 = uγ δsγ δ(u) +12uγ3kγ3 + u33K33+ u3L3. Using the definition ofL (see Section 5.1), it followsthatM11(U1)+M10(U1) = LM x and, therefore,M = (Id+L)M x. Choosing nowin (29) test functions of the form(Id +L)M , where

M = t((sαβ(v)

)α,β=1,2,

(kα3/2

)α=1,2, K33,02, L3

),(

v,(kα3)α=1,2, K33, L3

) ∈ VKL ×(L2(�)

)4,

keeping definition (24):RH = ∫Y(Id + tL)R(Id +L)dy in mind, we are led to

the weak formulation (34). This ends the proof. 2

6.3. STEP3: PLATE THEORY

We complete the proof of Theorem 5.1 by eliminatingkγ3,K33, N (L3), and pos-sibly, M(L3). The proof is based on the decompositionM = M0 ⊕M−1 ⊕M−2

where ally-terms are killed because everything in (34) depends only onx. Giventhat there is no confusion possible, we keep the same notation as before for thecorresponding functional spaces. We also use againM instead ofM x. In simplerterms, in the sequelM0,M−1,M−2, M designate

M−2 = {t (02×2,02,K33,03);K33 ∈ L2(�)},

M−1 = {t (02×2, (kα3/2)α=1,2,03, L3) with M(L3) = 0

for Dirichlet conditions},

M0 = {M00(v,0); v ∈ VKL

},

M = t(sαβ(u)α,β=1,2, (kα3/2)α=1,2,K33,02, L3

).

(35)


6.3.1. Case of Dirichlet Conditions

Here, we eliminate the componentskγ3,K33, L3, that is,5M , where5 is definedin (25). They are computed with respect toϕc (recall thatM(L3) = ϕc for Dirichletconditions) andM0 := M −5M . As usual in the plate theory, we are led to distin-guishM(M0) = t ((sγ δ(u))γ,δ=1,2,06) which contains the terms inu, andN (M0)

= t ((−x3∂2γ δu3)γ,δ=1,2

,06) which contains the terms inu3. Let us also recall the

definitionφc = t (09, ϕc).As for Dirichlet conditionsG = G1 = h = 0, problem (34) simply becomes∫

�

tMRHM dx = lu(v) ∀M ∈M, (36)

wherev is the vector ofVKL associated withM0. ChooseM ∈M(M−1⊕M−2) in(36). Using thatRH like R does not depend onx3, then∫

�

tMRHM dx =∫�

tMRHM(M)dx = 0 ∀M ∈M(M−1⊕M−2

).

Hence,RHM(M) ∈ (M(M−1 ⊕M−2))⊥. However, asRHM(M) = M(RHM)evidently belongs to(N (M−1⊕M−2))⊥, this is equivalent toRHM(M) ∈ (M−1⊕M−2)⊥. This implies52RM(M) = 0. Also, (36) implies that5RN (M) = 0.Using the decompositionM = 5M +M0, whereM0 ∈M0, we get

M(M) = 5M(M)+M(M0) = 52M(M)+M(M0)+ φc,N (M) = 5N (M)+N (M0).

Multiplying by52RH and5RH , respectively, we thus obtain

52RHM(M) = 0= (52R

H52)M(M)+52R

H(M(M0)+ φc

),

5RHN (M) = 0= (5RH5)N (M)+5RHN (M0),

or equivalently,

52M(M) = −(52RH52

)−152R

H(M(M0)+ φc

),

5N (M) = −(5RH5)−15RHN

(M0).

Finally, using definition (25) ofTN andTM andM(L3) = ϕc, we obtain

M(M) = (Id + TM)(M(M0)+ φc

),

N (M) = (Id + TN )N(M0).

Now, we choose in (36) test functionsM ∈M of the formM = (Id+TN )N (M0)+(Id+ TM)M(M0), whereM0 ∈M0. We get∫

�

(tM(M0)RMM

(M0)+ tN

(M0)RN N

(M0))

dx

= lu(v)−∫�1

tM(M0)RMφc dx ∀M0 ∈M0.


Noting that{M(M0) = ((sαβ(v))α,β=1,2,06

), N

(M0) = ((− x3∂

2αβv3

)α,β=1,2,06

),

M(M0) = ((sγ δ(u))γ,δ=1,2,06

), N

(M0) = ((− x3∂

2γ δu3

)γ,δ=1,2,06

),

(37)

this completes the proof of Theorem 5.1 for Dirichlet electrical boundary condi-tions.

6.3.2. Case of Local Mixed Conditions

Here, as in the Dirichlet conditions case, we eliminatekγ3,K33, andL3. We alsouse notation (35).

The variational formulation (34) is here reduced to∫�

(tMRHM + 2|Y1|GL3M(L3)

)dx = lu(v)+ lϕ

(M(L3)) ∀M ∈M. (38)

For test functions inM−1⊕M−2 one gets∫�

(tN(M)RHN (M)+ tM

(M)(

RH + 2|Y1|G51)M(M)

)dx

= |Y1|∫�

HM(5M)dx, (39)

where

H := t (0, h).

Arguing as in the case of Dirichlet conditions, (39) implies5RGM(M) = |Y1|Hand5RHN (M) = 0, whereRG = RH + 2|Y1|G51. Writing M = 5M +M0

with M0 ∈M0, one gets thenN (5M) =−(5RH5)−15RHN (M0) andM(5M)=−(5RG5)

−1 (5RH(M(M0)−|Y1|H). With definition (25) ofTN andTN thisis {

N (M) = (Id + TN )N(M0),

M(M) = (Id + TM)M(M0)+ |Y1|(5RG5)

−1H .(40)

For M0 ∈ M0, let us defineM ∈ M by: N (M) = (Id + TN )N (M0), M(M) =(Id + TM)M(M0). Then from (38) and (40):∫

�

(tM(M0)RMM

(M0)+ tN

(M0)RN N

(M0))dx = lu(v),

+ |Y1|∫�

tM(M0)(tTM −

(Id+tTM

)RG(5RG5)

−1)H dx ∀M0 ∈M0,

wherev is the vector ofVKL associated withM . With (37), this proves Theorem 5.1for local mixed conditions.


6.3.3. Case of Nonlocal Mixed Conditions

Here, onlykα3, K33, and N (L3) are eliminated. Because of the nonlocal termin G1, M(L3) cannot be eliminated. We also use notation (35). The variationalformulation of Lemma 6.5 is here∫

�

(MRHM + 2|Y1|GL3M(L3)

)dx + 2|Y1|G1

∫�

∂αM(L3)∂αL3 dx

= lu(v)+ lϕ(M(L3)) ∀M ∈M. (41)

For M ∈M−1⊕M−2, we thus obtain∫�

(tN(M)RN (M)+ tM

(M)(

R + 2|Y1|G51)M(M)

)dx

+ 2|Y1|G1

∫�

∂αM(L3)∂αL3dx = 0. (42)

With the decompositionM = 5M + M0, we deduce as beforeN (5M) =−(5RH5)−1N (M0). ConcerningM(M), we use the decompositionM = 51M+52M +M0. With test functionsM ∈M(52M) in (42), we therefore getM(52M)=−(52R

H52)−1(M(M0)+M(52M). This leads to{

N (M) = (Id + TN )N(M0),

M(M) = (Id + TM)M(M0+33

).

(43)

With the appropriate choice of test functions in (41) the weak formulation ofTheorem 5.1 (ii) follows as in Sections 6.3.1 and 6.3.2.

7. Example

In this section, we propose quite explicit formulae for operators and effective coef-ficients described in Section 5. We consider the particular case of a plate made upof transversally isotropic material with Dirichlet electrical boundary conditions.

7.1. PRELIMINARIES

The projections5,51 and52 are:

5 =

02×2×2×2

02×2×2

02×2

02×2×2

02×2

02×2×2

(δα,µ)α,µ=1,2

02

02×2

02

02×2

02

102

0

02×2×2

02×2

02

02×2

02

02×2

02

002

1

,

51 =

02×2×2×2

02×2×2

02×2

02×2×2

02×2

02×2×2

02×2

02

02×2

02

02×2

02

002

0

02×2×2

02×2

02

02×2

02

02×2

02

002

1

,


52 =

02×2×2×2

02×2×2

02×2

02×2×2

02×2

02×2×2

(δα,µ)α,µ=1,2

02

02×2

02

02×2

02

102

0

02×2×2

02×2

02

02×2

02

02×2

02

002

0

.Then,

5R =

02×2×2×2 02×2×2 02×2

(Rα3γ δ)α,γ,δ=1,2 (4Rα3γ3)α,γ=1,2 (2Rα333)α=1,2

(R33γ δ)γ,δ=1,2 (2R33γ3)γ=1,2 R3333

02×2×2 02×2 02

(−d3γ δ)γ,δ=1,2 (−2d3γ3)3,γ=1,2 −d333

02×2×2 02×2

(2dγα3)α,γ=1,2 (2d3α3)α=1,2

(dγ33)γ=1,2 d333

02×2 02

(c3γ )γ=1,2 c33

and

5R5 =

02×2×2×2 02×2×2 02×2 02×2×2 02×2

02×2×2 (4Rα3γ3)α,γ=1,2 (2Rα333)α=1,2 02×2 (2d3α3)α=1,2

02×2 (2R33γ3)γ=1,2 R3333 02 d333

02×2×2 02×2 02 02×2 02

02×2 (−2d3γ3)3,γ=1,2 −d333 02 c33

.The matrix (5R5)−1 is computed by inverting the above matrix, omitting therows and columns of zeros, that is: (4Rα3γ3)α,γ=1,2 (2Rα333)α=1,2 (2d3α3)α=1,2

(2R33γ3)γ=1,2 R3333 d333

(−2d3γ3)3,γ=1,2 −d333 c33

.The result of inversion is then replaced in a 10× 10 matrix.

With this, the computation ofTN = −(5RH5)−15RH is clear. The computa-tion of TM is similar, and the effective coefficients follow. We go into detail aboutthese last computations in the next subsection.

7.2. EXPLICIT FORMULAE FOR A TRANSVERSALLY ISOTROPIC MATERIAL

WITH DIRICHLET CONDITIONS

The first step consists in computing the effective homogenized coefficientsRH .This cannot be done analytically, but can be obtained by standard numerical com-putation. Here, we assume thatRH is known and has the same form of isotropyasR. We compute the stiffness tensor of the two-dimensional plate model. Forsimplicity, the indexH on the coefficient ofRH has been removed.


For a transversally isotropic material, the global stiffness-piezoelectricity-permi-tivity tensor

R =

R1111R1211R2111R22112R13112R2311R3311−d111−d211−d311

R1112R1212R2112R22122R13122R2312R3312−d112−d212−d312

R1121R1221R2121R22212R13212R2321R3321−d121−d221−d321

R1122R1222R2122R22222R13222R2322R3322−d122−d222−d322

2R11132R12132R21132R22134R13134R2313R3313−2d113−2d213−2d313

2R11232R12232R21232R22234R13234R2323R3323−2d123−2d223−2d323

R1133R1233R2133R2233R1333R2333R3333−d133−d233−d333

d111d112d121d1222d1132d123d133c11c21c31

d211d212d221d2222d2132d223d233c12c22c31

d311d312d221d3222d3132d323d333c13c23c33

reduces to

C11 0 0 C12 00 (C11− C12)/2 0 0 00 0 (C11− C12)/2 0 0C12 0 0 C11 00 0 0 0 2(C11− C12)0 0 0 0 0C12 0 0 0 00 0 0 0 −2E150 0 0 0 0−E31 0 0 −E31 0

0 C12 0 0 E310 0 0 0 00 0 0 0 00 C12 0 0 E310 0 2E15 0 0

2(C11− C12) 0 0 2E24 00 C11 0 0 E330 0 ε11 0 0

−2E24 0 0 ε11 00 −E33 0 0 ε22


whereC11, C12, E31, E24, E15, ε11, andε22 are fixed positive constants. Applicationof the formulas to Dirichlet conditions:RN = (Id+ tTN )R(Id+TN ) andRM =(Id + tTM)R(Id + TM) gives the expression of the stiffness coefficients of thetwo-dimensional model:

RM1111

RM1211

RM2111

RM2211

RM1112

RM1212

RM2112

RM2212

RM1121

RM1221

RM2121

RM2221

RM1122

RM1222

RM2122

RM2222

=

C11− C2

12/C11 0 0 C12

0 (C11− C12)/2 0 00 0 (C11− C12)/2 0

C12− C212/C11 0 0 C11

,and

RN 1111

RN 1211

RN 2111

RN 2211

RN 1112

RN 1212

RN 2112

RN 2212

RN 1121

RN 1221

RN 2121

RN 2221

RN 1122

RN 1222

RN 2122

RN 2222

=

RN 1111 0 0 RN 1122

0 (C11− C12)/2 0 00 0 (C11− C12)/2 0

RN 2211 0 0 RN 2222

,where

RN 1111 =((C2

12E233ε22+ E2

31ε322+ C11E33

(2C2

12E33+ E231E33+ E3

33

− 2C12E31ε22)− 2C2

11

(− C12E31E33+ C212ε22+ E2

33ε22)

+ 2C12E31E33(E2

33− ε222

)+ C311

(C2

12+ ε222

))/(E2

33− C11ε22)2,

RN 1122 =(E2

31ε322+ C11E33

(E2

31E33− C12E31ε22− 2C12E33ε22)

+ C211C12

(E31E33+ ε2

22

)+ C12(− E31E

333+ E4

33− E31E33ε222

))/(E2

33− C11ε22)2,

RN 2211 =(E2

31ε322+ C11E33

(C2

12E33+ E231E33− 2C12(E31+ E33)ε22

)+ C12

(4E31E

333+ E4

33− E31E33ε222

)+ C211C12(E31E33

+ ε22(−C12+ ε22)))/(

E233− C11ε22

)2,

RN 2222 =(− 2C2

11E233ε22+ C3

11ε222+ C11E33

(E2

31E33+ E333− C12E31ε22

)+ E31

(C12E

333+ E31ε

322

))/(E2

33− C11ε22)2).


The piezoelectric coefficients for the 2-dimensional plate model are given bydM311

dM312

dM321

dM322

=E31− (C12E33)/C11

00

E31− (C12E33)/C11

.REMARK 7.1.

(i) As compared with the complexity of these formulae, the formulation of Sec-tion 5 is very synthetic.

(ii) The above computations have been carried out with Mathematica.(iii) For multilayered 2-dimensional plate models the results are much more com-

plicated. Using our approach, the complexity is the same.

References

1. G. Allaire, Homogenization and two-scale convergence.SIAM J. Math. Anal.23(26) (1992)1482–1518.

2. A. Bensoussan, J.L. Lions and G. Papanicolaou,Asymptotic Analysis for Periodic Structures.North-Holland, Amsterdam (1978).

3. D. Caillerie, Thin elastic and periodic plates.Math. Methods Appl. Sci.6 (1984) 159–191.4. É. Canon and M. Lenczner, Models of elastic plates with piezoelectric inclusions, Part I:

Models without homogenization.Math. Comput. Modelling26(5) (1997) 79–106.5. É. Canon and M. Lenczner, Deux modèles de plaque mince avec inclusions piézoélectriques et

circuits électroniques distribués.C.R. Acad. Sci. Paris Sér. II326(1998) 793–798.6. P.G. Ciarlet,Mathematical Elasticity, Vol. II: Theory of Plates. North-Holland, Amsterdam

(1997).7. P.G. Ciarlet and P. Destuynder, A justification of the two-dimensional plate model.Comput.

Methods Appl. Mech. Engrg.18 (1979) 227–258.8. A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows.SIAM J.

Appl. Math.54(2) (1994) 333–408.9. G. Nguentseng, A general convergence result for a functional related to the theory of

homogenization.SIAM J. Math. Anal.20(3) (1989) 608–623.10. A. Raoult, Contribution à l’étude des modèles d’évolution de plaques et à l’approximation

d’équations d’évolution linéaires du second ordre par des méthodes multi-pas, thèse de 3èmecycle, Université Pierre et Marie Curie, Paris (1980).

11. E. Sanchez-Palencia,Non-Homogeneous Media and Vibration Theory. Lecture Notes inPhysics 127. Springer, Berlin (1980).

12. A.H. Zemanian,Infinite Electrical Networks. Cambridge Univ. Press, New York (1991).13. A.H. Zemanian, Transfinite graphs and electrical networks.Trans. Amer. Math. Soc.334(1992)

1–36.

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